How to have rounded (beveled?) edges in inset?
You can effectively do it extruding first.


Inset I
Extrude inward E
Select the inner edges
Bevel CtrlB and tune the amount of bevel edges with the mouse wheel or with the numpad + and -


In 2.79 (as you have here)



Or in 2.8 you can also use the new tools (keyboard short cut are still available, though)